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I wrote up the proof for myself about 4 years ago, here it is. Definition. Fix a universal Turing machine with one program tape and one input tape. Assume that the possible programs are binary number …

I am not sure what you mean by "closed formula", but I provide below an identity that might satisfy you. It is a consequence of Gauss's lemma and it implies quadratic reciprocity (for details see Theo …

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It is known that $f(n):=r_2(n)/4$ is a multiplicative function such that for $p\equiv 1\pmod{4}$ we have $f(p^k)=k+1$, while for $p\equiv 3\pmod{4}$ we have $f(p^k)=1$ or $f(p^k)=0$ depending on wheth …

(13) follows easily by induction, namely it is true for $m=0$ and $m=1$ by inspection, and then the recursion yields it for all $m$. (11) and (12) hold more generally for all $k$ (including the odd on …

According to Dickson (History of numbers Vol. 2, Ch. XII, p.376), Göpel (Jour. für Math. 45, 1853, 1-14) proved your conjectures "by use of continued fractions". Actually Jour. für Math. stands for C …

If you want to do research in the circle method, then my best advice is to study Vaughan's book as it is the definite source on the subject. I have studied it by myself (as an undergraduate) and soon …

See Theorem 9.10 in Montgomery-Vaughan: Multiplicative number theory I (Cambridge University Press, 2006).

This is a supplement to Jeremy Rouse's nice answer. Let $\alpha$ and $\beta$ be positive irrational numbers. Skolem proved in 1957 (see Theorem 8 in On certain distributions of integers in pairs with …

This is a revised version of my original answer. I fixed some inconsistencies and made the text more readable. The correct statement is given by the deterministic Miller-Rabin test coupled with an es …

I agree with Felipe Voloch that this question is not quite suitable for MO. At any rate, it is well-known (folklore) among number theorists that solving any single $k=2$ case of Dickson's conjecture w …

This statement is proved in detail in Tenenbaum's book "Introduction to analytic and probabilistic number theory". See Theorem 2 in Section III.1.2. See also Theorem 3 in Section III.1.3, where it is …

You ask if Waring's conjecture has been proved. The answer is yes (Hilbert 1909), and you can read about its history here. Your $k(p)$ is usually denoted by $g(p)$, and its minimal value is almost p …

Your expectation is correct. Let $\lambda_{j} \leq U < \lambda_{j+1}$. There are independent lattice vectors $v_1,\dots,v_j\in\Lambda$ such that $\|v_i\|=\lambda_i$ for $1\leq i\leq j$. Consider the …

See Theorem 4.1.11 and its proof in Niederreiter-Xing: Algebraic geometry in coding theory and cryptography. Alternately, one can observe that the Riemann-Roch theorem in this setting is equivalent t …